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Semi-blind two-way AF relaying over Nakagami-m fading environment

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Abstract

In this paper, we investigate semi-blind opportunistic amplify-and-forward (AF) selection relaying in two-way dual-hop cooperative communication networks on independent Nakagami-m fading channels. This system is compared to channel state information (CSI)-assisted opportunistic AF selection relaying (CSIA-OAF) where the relays use variable gains for the amplification. In semi-blind opportunistic AF scenario (SB-OAF), the relays use partial CSI-fixed gains to amplify the received signals. In this work, we first derive the expression of the signal-to-noise ratio (SNR) in the case of SB-OAF selection relaying. The obtained SNR is used to process the bounds of average sum-rate, outage probability, and average symbol error rate (SER). Numerical results are used to show the performance of the proposed SB-OAF system compared to the CSIA-OAF relaying. The comparison shows that the cost of the significant reduction of the proposed SB-OAF complexity is obtained for a slight loss in performance compared to CSIA-OAF scenario.

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Authors and Affiliations

  1. SYSCOM LAB, ENIT, El Manar University, Tunis, Tunisia

    Wided Hadj Alouane & Noureddine Hamdi

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  1. Wided Hadj Alouane

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  2. Noureddine Hamdi

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Correspondence toWided Hadj Alouane.

Appendices

Appendix A

\(\frac {dF_{x}}{dx}\) is the convolution of\(f_{P\gamma _{N_{1}R_{i}}}\) and\(f_{P\gamma _{R_{i}N_{2}}}\) which can be expressed as follows:

$$ \frac{dF_{x}}{dx}=\int_{0}^{x}f_{P\gamma_{N_{1}R_{i}}}(\beta)f_{P\gamma_{R_{i}N_{2}}}(x-\beta) d\beta $$
(37)

By substituting Eqs. 11 and12 into Eq. 37, we get

$$\begin{array}{@{}rcl@{}} \frac{dF_{x}}{dx} &=& \left(\frac{m_{1i}}{P\overline{\gamma}_{N_{1}R_{i}}}\right)^{m_{1i}}\left(\frac{m_{2i}}{P\overline{\gamma}_{R_{i}N_{2}}}\right)^{m_{2i}}\\ &&{\kern12pt}\times\frac{1}{{\Gamma}(m_{1i}) {\Gamma}\left(m_{2i}\right)} \exp\left(\frac{-m_{2i}x}{P\overline{\gamma}_{R_{i}N_{2}}}\right) \\ &&\sum\limits_{k=0}^{m_{2i}-1} \left(\begin{array}{c}{m_{2i}-1}\\ {k}\end{array}\right) x^{m_{2i}-1-k} (-1)^{k}\int_{0}^{x}(\beta)^{\left(m_{1i}-1+k\right)} \\ &&\exp\left(-\beta\left(\frac{m_{1i}}{P\overline{\gamma}_{N_{1}R_{i}}}-\frac{m_{2i}}{P\overline{\gamma}_{R_{i}N_{2}}}\right) \right) d\beta \end{array} $$
(38)

The above integral can be expressed as follows:

$$\begin{array}{@{}rcl@{}} &&\int_{0}^{x}(\beta)^{\left(m_{1i}-1+k \right)} \exp\left(-\beta\left(\frac{m_{1i}}{P\overline{\gamma}_{N_{1}R_{i}}}- \frac{m_{2i}}{P\overline{\gamma}_{R_{i}N_{2}}}\right) \right) d\beta \\ &&=\left(\frac{m_{1i}\overline{\gamma}_{R_{i}N_{2}}-m_{2i}\overline{\gamma}_{N_{1}R_{i}}}{P\overline{\gamma}_{N_{1}R_{i}}\overline{\gamma}_{R_{i}N_{2}}}\right)^{-k-m_{1i}} \\ &&\left({\Gamma}\left(k+m_{1i}\right)-{\Gamma}\left(k+m_{1i},x\left(\frac{m_{1i}\overline{\gamma}_{R_{i}N_{2}}-m_{2i}\overline{\gamma}_{N_{1}R_{i}}}{P\overline{\gamma}_{N_{1}R_{i}}\overline{\gamma}_{R_{i}N_{2}}}\right)\right)\right) \end{array} $$
(39)

By substituting Eq. 39 into Eq. 38, we have

$$\begin{array}{@{}rcl@{}} \frac{dF_{x}}{dx}&=& \left(\frac{m_{1i}}{P\overline{\gamma}_{N_{1}R_{i}}}\right)^{m_{1i}}\left(\frac{m_{2i}}{P\overline{\gamma}_{R_{i}N_{2}}}\right)^{m_{2i}} \\&&{\kern12pt} \times\frac{1}{{\Gamma}\left(m_{1i}\right){\Gamma}\left(m_{2i}\right)}\exp\left(\frac{-m_{2i}x}{P\overline{\gamma}_{R_{i}N_{2}}} \right)\\&&\sum\limits_{k=0}^{m_{2i}-1}\left(\begin{array}{c}{m_{2i}-1}\\ {k}\end{array}\right)(-1)^{k} x^{m_{2i}-1-k}\\ &&{\kern1pc} \times\left(\frac{m_{1i}\overline{\gamma}_{R_{i}N_{2}}-m_{2i}\overline{\gamma}_{N_{1}R_{i}}}{P\overline{\gamma}_{N_{1}R_{i}}\overline{\gamma}_{R_{i}N_{2}}}\right)^{-k-m_{1i}} \\&&\left({\Gamma}{\vphantom{\left.\left(k+m_{1i},x\left(\frac{m_{1i}\overline{\gamma}_{R_{i}N_{2}}-m_{2i}\overline{\gamma}_{N_{1}R_{i}}}{P\overline{\gamma}_{N_{1}R_{i}}\overline{\gamma}_{R_{i}N_{2}}}\right)\right)\right)}}\left(k+m_{1i}\right)-{\Gamma} \left(k+m_{1i},x\left(\frac{m_{1i}\overline{\gamma}_{R_{i}N_{2}}-m_{2i}\overline{\gamma}_{N_{1}R_{i}}}{P\overline{\gamma}_{N_{1}R_{i}}\overline{\gamma}_{R_{i}N_{2}}}\right)\right)\right) \end{array} $$
(40)

Applying the finite-sum representation of the incomplete Gamma function given by Γ(α,x) = (α−1)!\(\exp (-x) \sum _{l=0}^{\alpha -1} \frac {1}{l!}x^{l},\) Eq. 40 can be rewritten as Eq. 13.

Appendix B

The upper bound for the average sum-rate of SB-OAF system over Nakagami-m fading channels is written as [32]

$$\begin{array}{@{}rcl@{}} &&{}E\left\{R_{SB-OAF}^{UB}\right\}=\frac{1}{2ln(2)}\\ &&{\kern4.5pc} \times\int_{0}^{\infty} ln\left(\frac{\sqrt{P}\;P_{R_{i}}(C-1)\sqrt{C-1}(x)^{\frac{3}{2}}}{4C}\right)f_{\gamma}(x) dx \end{array} $$
(41)

Wherefγ(x) is the PDF ofγ given asγ =max (γ1,...,γL). Equation 41 can be rewritten after simple manipulations as follows:

$$\begin{array}{@{}rcl@{}} E\left\{R_{SB-OAF}^{UB} \right\} &=&\frac{1}{2ln(2)}\int_{0}^{\infty} ln\left( \frac{\sqrt{P}\;P_{R_{i}}(C-1) \sqrt{C-1}\;t}{4C}\right)\frac{2f_{\gamma}\left(t^{\frac{2}{3}}\right)}{3t^{\frac{1}{3}}} dt \\ &=&\frac{L}{2ln(2)}\int_{0}^{\infty}\left(ln\left(\frac{\sqrt{P}\;P_{R_{i}}(C-1)\sqrt{C-1}}{4C}\right)+ln(t)\right)\\ &&\left({\Gamma}\left(m,\frac{m t^{\frac{2}{3}}}{\overline{\gamma}_{b}} \right)\right)^{L-1}\left(\frac{m}{\overline{\gamma}_{b}}\right)^{m} t^{\frac{2m}{3}-1} \frac{2\exp\left(\frac{-mt^{\frac{2}{3}}}{\overline{\gamma}_{b}}\right)}{3 {\Gamma}(m)} dt \end{array} $$
(42)

When the Nakagami fading parameter m is an integer value, Γ(m) = (m−1)! and using the definition of Γ(.,.) given in the second line after Eq. 21, we can simply show that

$$\begin{array}{@{}rcl@{}} {\Gamma}\left(m,\frac{m t^{\frac{2}{3}}}{\overline{\gamma}_{b}}\right)&=&\frac{1}{(m-1)!}\int_{0}^{\frac{m t^{\frac{2}{3}}}{\overline{\gamma}_{b}}} t^{m-1}\exp(-t) dt \\ &=&1-\exp\left(\frac{-m t^{\frac{2}{3}}}{\overline{\gamma}_{b}}\right)\sum\limits_{l=0}^{m}\frac{1}{l!}\left(\frac{m t^{\frac{2}{3}}}{\overline{\gamma}_{b}}\right)^{l} \end{array} $$
(43)

Using Eq. 22, the expansion\(\left ({\Gamma }\left (m,\frac {m t^{\frac {2}{3}}}{\overline {\gamma }_{b}}\right )\right )^{L-1}\) in Eq. 42 can be written in terms of finite series expansions as follows:

$$\begin{array}{@{}rcl@{}} \left({\Gamma}\left(m,\frac{m t^{\frac{2}{3}}}{\overline{\gamma}_{b}}\right)\right)^{L-1}&=&\left(1-\exp\left(\frac{-m t^{\frac{2}{3}}}{\overline{\gamma}_{b}} \right) \sum\limits_{l=0}^{m}\frac{1}{l!} \left( \frac{m t^{\frac{2}{3}}}{\overline{\gamma}_{b}}\right)^{l} \right)^{L-1} \\ &=&\sum\limits_{i=0}^{L-1} \left(\begin{array}{c}{L-1}\\ {i}\end{array}\right)(-1)^{i} \exp\left(\frac{-im t^{\frac{2}{3}}}{\overline{\gamma}_{b}}\right) \\&&{\kern9pt}\times\left(\sum\limits_{l=0}^{m}\frac{1}{l!}\left(\frac{m t^{\frac{2}{3}}}{\overline{\gamma}_{b}}\right)^{l}\right)^{i}\\ &=&\sum\limits_{i=0}^{L-1} \sum\limits_{z=0}^{i(m-1)} \left(\begin{array}{c}{L-1}\\ {i}\end{array}\right) (-1)^{i} a_{i,z}\\&&{\kern8pt} \times \exp\left(\frac{-im t^{\frac{2}{3}}}{\overline{\gamma}_{b}}\right)\left(\frac{m t^{\frac{2}{3}}}{\overline{\gamma}_{b}}\right)^{z} \end{array} $$
(44)

Where the expression in the second line of Eq. 44 is simplified using the expansion

$$ \sum\limits_{l=0}^{m}\frac{1}{l!} \left(\frac{m t^{\frac{2}{3}}}{\overline{\gamma}_{b}}\right)^{l}=\sum\limits_{z=0}^{i(m-1)} a_{i,z} \left(\frac{m t^{\frac{2}{3}}}{\overline{\gamma}_{b}}\right)^{z} $$
(45)

Where the coefficientsai,z are defined in Eq. 18. Substituting the expression Eq. 44 into Eq. 42, and after some manipulations, we obtain

$$\begin{array}{@{}rcl@{}} E\left\{R_{SB-OAF}^{UB}\right\} &=&\frac{L}{3ln(2)} \sum\limits_{i=0}^{L-1} \sum\limits_{z=0}^{i(m-1)} \left(\begin{array}{c}{L-1}\\ {i}\end{array}\right)(-1)^{i}a_{i,z} \left(\frac{m}{\overline{\gamma}_{b}}\right)^{m+z}\frac{1}{{\Gamma}(m)}\\&&\left(\int_{0}^{\infty}ln\left(\frac{\sqrt{P}\;P_{R_{i}}(C-1)\sqrt{C-1}}{4C}\right)t^{\frac{2}{3}(m+z)-1}\right.\\ &&{\kern4pt}\times\exp\left(\frac{-mt^{\frac{2}{3}}(1+i)}{\overline{\gamma}_{b}}\right)dt\\&& \left.+\int_{0}^{\infty}ln(t)t^{\frac{2}{3}(m+z)-1} \exp\left(\frac{-m t^{\frac{2}{3}}(1+i)}{\overline{\gamma}_{b}}\right)dt\right) \end{array} $$
(46)

The above integrals can be written as

$$\begin{array}{@{}rcl@{}} &&{} \int_{0}^{\infty} t^{\frac{2}{3}(m+z)-1} \exp\left(\frac{-m t^{\frac{2}{3}}(1+i)}{\overline{\gamma}_{b}}\right)dt\\ &&{\kern12pt}=\frac{3}{2}(1+i)^{-m-z}\left(\frac{m}{\overline{\gamma}_{b}}\right)^{-m-z} {\Gamma}( m+z) \end{array} $$
(47)

and

$$\begin{array}{@{}rcl@{}} &&{}\int_{0}^{\infty} ln(t) t^{\frac{2}{3}(m+z)-1} \exp\left(\frac{-m t^{\frac{2}{3}}(1+i) }{\overline{\gamma}_{b}}\right)dt \\ &&{}=\frac{9}{4}(1+i)^{-m-z}\left(\frac{m}{\overline{\gamma}_{b}}\right)^{-m-z}{\Gamma}(m+z)\\ &&{\kern18pt}\times\left(PG[0,m+z]-ln\left(\frac{m(1+i)}{\overline{\gamma}_{b}} \right)\right) \end{array} $$
(48)

Substituting Eqs. 47 and48 into Eq. 46, and after some manipulations, we obtain Eq. 17.

Appendix C

The upper bound for the average sum-rate of two-way system with CSI-assisted relays is written as in [31]

$$ E\left\{R_{CSIA-OAF}^{UB} \right\} =\frac{1}{2ln(2)}\int_{0}^{\infty} ln\left(\frac{P_{R_{i}}^{2}}{\lambda_{i}}x^{2}\right) f_{\gamma}(x) dx $$
(49)

Wherefγ(x) is the PDF ofγ given asγ =max (γ1,...,γL). Considering Nakagami-m fading channels, Eq. 49 can be rewritten after simple manipulations as follows:

$$\begin{array}{@{}rcl@{}} E\left\{R_{CSIA-OAF}^{UB}\right\} &=&\frac{1}{2ln(2)}\int_{0}^{\infty} ln\left(\frac{P_{R_{i}}^{2}}{\lambda_{i}}\; t \right)\frac{f_{\gamma}\left(\sqrt{t}\right)}{2\sqrt{t}} dt \\ &=&\frac{L}{4ln(2)}\int_{0}^{\infty}\left(ln\left(\frac{P_{R_{i}}^{2}}{\lambda_{i}}\right)+ln(t) \right)\\&&\times\left({\Gamma}\left(m,\frac{m\sqrt{t}}{\overline{\gamma}_{b}}\right)\right)^{L-1} \\&&{} \left(\frac{m}{\overline{\gamma}_{b}}\right)^{m}\times t^{\frac{m}{2}-1} \frac{\exp\left(\frac{-m \sqrt{t}}{\overline{\gamma}_{b}}\right)}{{\Gamma}(m)}dt \end{array} $$
(50)

The expansion\(\left ({\Gamma }\left (m,\frac {m\sqrt {t}}{\overline {\gamma }_{b}}\right )\right )^{L-1}\) can be written using Eq. 44 as follows:

$$\begin{array}{@{}rcl@{}} &&{}\left({\Gamma}\left(m,\frac{m \sqrt{t}}{\overline{\gamma}_{b}}\right)\right)^{L-1}=\sum\limits_{i=0}^{L-1}\sum\limits_{z=0}^{i(m-1)} \left(\begin{array}{c}{L-1}\\ {i}\end{array}\right)(-1)^{i}a_{i,z}\\ &&{\kern12pt} \times\exp\left(\frac{-im\sqrt{t}}{\overline{\gamma}_{b}}\right)\left(\frac{m\sqrt{t}}{\overline{\gamma}_{b}} \right)^{z} \end{array} $$
(51)

Substituting Eq. 51 into Eq. 50, and after some manipulations, we obtain

$$\begin{array}{@{}rcl@{}} E\left\{R_{SB-OAF}^{UB}\right\}&=&\frac{L}{4ln(2)} \sum\limits_{i=0}^{L-1} \sum\limits_{z=0}^{i(m-1)}\left(\begin{array}{c}{L-1}\\ {i}\end{array}\right)(-1)^{i} a_{i,z} \\ &&{\kern12pt} \times \left(\frac{m}{\overline{\gamma}_{b}}\right)^{m+z}\frac{1}{{\Gamma}(m)}\\&& \left(\int_{0}^{\infty}ln\left(\frac{P_{R_{i}}^{2}}{\lambda_{i}}\right) t^{\frac{1}{2}(m+z)-1} \exp\left(\frac{-m\sqrt{t}(1+i)}{\overline{\gamma}_{b}}\right)dt\right.\\&& \left.+\int_{0}^{\infty}ln(t)t^{\frac{1}{2}(m+z)-1}\exp\left(\frac{-m \sqrt{t}(1+i)}{\overline{\gamma}_{b}}\right)dt\right) \end{array} $$
(52)

The above integrals can be written as

$$\begin{array}{@{}rcl@{}} &&\int_{0}^{\infty} t^{\frac{1}{2}(m+z)-1}\exp\left(\frac{-m\sqrt{t}(1+i)}{\overline{\gamma}_{b}}\right)dt \\&&{\kern5pc} \times=2(1+i)^{-m-z}\left(\frac{m}{\overline{\gamma}_{b}}\right)^{-m-z}{\Gamma}(m+z) \end{array} $$
(53)

and

$$\begin{array}{@{}rcl@{}} &&{}\int_{0}^{\infty} ln(t)t^{\frac{1}{2}(m+z)-1}\exp\left(\frac{-m \sqrt{t}(1+i)}{\overline{\gamma}_{b}}\right)dt \\ &&{\kern5pc} =4 (1+i)^{-m-z} \left(\frac{m}{\overline{\gamma}_{b}}\right)^{-m-z} {\Gamma}(m+z) \\&&{\kern6pc}\left(PG[0,m+z]-ln\left(\frac{m(1+i)}{\overline{\gamma}_{b}}\right)\right) \end{array} $$
(54)

Substituting Eqs. 53 and54 into Eq. 52, and after some manipulations, we obtain Eq. 19.

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Hadj Alouane, W., Hamdi, N. Semi-blind two-way AF relaying over Nakagami-m fading environment.Ann. Telecommun.70, 49–62 (2015). https://doi.org/10.1007/s12243-014-0427-6

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